1. Field of the Invention
The present invention relates to a signal processing device for estimating a correlation matrix in such a manner to adapt to input signals.
2. Description of the Related Art
Correlation matrix estimation is used for MUSIC, BF (BeamFormer), or BSS (Blind Source Separation) (see N. Kikuma, Adaptive Signal Processing with Array Antenna, Kagaku Gijutsu Shuppan, Inc., 1999, and H. Saruwatari et al., IEEE Trans. on Speech and Audio Process, vol. 14(2), pp. 666-678, 2006). The correlation matrix Rxx is defined by Equation (1) for signal x(t) represented as an Nth-dimensional column vector using discrete time t as a variable.Rxx=E[x(t)xH(t)],  (1)where H denotes a complex conjugate transposed matrix, and E(A(t)) represents an expected value or time average for A(t). Since Equation (1) includes the expected value operation, the calculation of the correlation matrix requires that the signal x(t) be known at all times t.
However, since the adaptive BF or adaptive BSS cannot use future signals and the signal varies over time, signals acquired up to the time of buffering are used to estimate the correlation matrix. Specifically, an estimated correlation matrix Rxx^(t) at time t is calculated using a window function w(t) for signal extraction according to Equation (2).
                                                                                          R                  xx                  ^                                ⁡                                  (                  t                  )                                            =                            ⁢                                                w                  ⁡                                      (                    t                    )                                                  *                                  [                                                            x                      ⁡                                              (                        t                        )                                                              ⁢                                                                  x                        H                                            ⁡                                              (                        t                        )                                                                              ]                                                                                                                        =                                ⁢                                                      ∑                    τ                                    ⁢                                                                          ⁢                                                            w                      ⁡                                              (                        τ                        )                                                              ·                                          [                                                                        x                          ⁡                                                      (                                                          t                              -                              τ                                                        )                                                                          ⁢                                                                              x                            H                                                    ⁡                                                      (                                                          t                              -                              τ                                                        )                                                                                              ]                                                                                  ,                                                          (        2        )            where * represents convolution. In the actual processing, a rectangular window having a certain length is often used as the window function w(t). In order to reduce the amount of calculation, a technique for calculating a spaced, estimated correlation matrix Rxx^(t) and a technique using calculated values based on only data in the current time have been proposed (see J. M. Valin et al., Proc. IEEE/RSJ Intelligent Robot and Systems, pp. 2123-2128, 2004, and Nakajima et al., Technical Report of IEICE, Vol. EA2007-30, pp. 19-24, 2007).
The accuracy of correlation matrix estimation varies depending on the kind of window. Here, two signals x1(t) and x2(t) are considered, which are respectively defined by Equations (3) and (4) in which the correlation value r is a (|a|≦1). For the sake of simplicity, signal power is normalized.x1(t)=n1(t),  (3)x2(t)=an1(t)+(1−a2)1/2n2(t),  (4)where n1(t) and n2(t) represent noise signals that follow the standard normal distribution. If the expected value operation using a rectangular window having length N is defined by Equation (5), the estimated correlation value r^ of the two signals is calculated according to Equation (7).EN[u(t)]=(1/N)Σt=0-N-1u(t)  (5)
                                                                        r                ^                            =                            ⁢                                                E                  N                                ⁡                                  [                                                                                    x                        1                                            ⁡                                              (                        t                        )                                                              ⁢                                                                  x                        2                                            ⁡                                              (                        t                        )                                                                              ]                                                                                                        =                            ⁢                                                a                  ⁢                                                                          ⁢                                                            E                      N                                        ⁡                                          [                                                                        n                          1                          2                                                ⁡                                                  (                          t                          )                                                                    ]                                                                      +                                                                            (                                              1                        -                                                  a                          2                                                                    )                                                              1                      /                      2                                                        ⁢                                                            E                      N                                        ⁡                                          [                                                                                                    n                            1                                                    ⁡                                                      (                            t                            )                                                                          ⁢                                                                              n                            2                                                    ⁡                                                      (                            t                            )                                                                                              ]                                                                                                                              (        7        )            
As N approaches infinity, since EN[n12(t)] approaches 1 whereas EN[n1(t)n2(t)] approaches 0, the estimated correlation value r^ substantially matches a true value. However, if N is finite, an error e expressed by Equation (8) occurs.
                                                        e              =                            ⁢                                                r                  ^                                -                r                                                                                        =                            ⁢                                                a                  ⁡                                      (                                                                                            E                          N                                                ⁡                                                  [                                                                                    n                              1                              2                                                        ⁡                                                          (                              t                              )                                                                                ]                                                                    -                      1                                        )                                                  +                                                                            (                                              1                        -                                                  a                          2                                                                    )                                                              1                      /                      2                                                        ⁢                                                            E                      N                                        ⁡                                          [                                                                                                    n                            1                                                    ⁡                                                      (                            t                            )                                                                          ⁢                                                                              n                            2                                                    ⁡                                                      (                            t                            )                                                                                              ]                                                                                                                              (        8        )            
EN[n12(t)] follows the chi-square distribution χN2 with a N degrees of freedom, and its mean is 1 and the dispersion is 2/N. EN[n1(t)n2(t)] follows the product of normal distribution, and it is estimated that its mean is 0 and the dispersion is 1/N. It shows that the average power E[e2] of the estimated error of the correlation value is inversely proportional to N as expressed in Equation (9).E[e2]=(a2+1)/N  (9)
FIG. 7 illustrates the calculation results of RMS (Root Mean Square) error values when uncorrelated signals are estimated using the rectangular window. The experimental values (indicated by the x marks) were calculated from the trials of 1000 times based on random numbers that follow two independent Gaussian distributions. The theoretical values (indicated by the dotted line) were set to 1/N1/2. It is apparent from FIG. 7 that the theoretical values and the experimental values match each other. The correlation value of the signals does not become 0 even if the signals are uncorrelated signals, and the RMS values are proportional to N−1/2. FIG. 8 illustrates RMS error values when the correlation value a has varied on condition that the window length N was fixed to 1000. It is apparent from FIG. 8 that the error increases as the correlation value a becomes large. Note that although the real signals were treated in this analysis, the same analysis can be applied to a correlation value EN[z1*(t)z2(t)] of complex signals z1(t) and z2(t), in each of which the real part and the imaginary part of the values follow the independent Gaussian distributions, to show that the error average power is proportional to N−1/2. In this case, however, since the dispersion of EN[|n1(t)|2] becomes 1/N, the error average power becomes constant regardless of the correlation value a.
When a correlation value is estimated for input signals discretely or continuously, an exponential window exhibits better effects than the rectangular window in terms of the amount of storage and the amount of calculation. The exponential window is often used for signal power estimation (see I. Kohen and B. Berdugo, Signal Processing, Vol. 81, 2001, pp. 2403-2418, 2001). On the other hand, there are fewer reports for correlation matrix estimation.
The following describes the fact that the estimation accuracy of the estimate value depends on the area of a squared window. When the expected value operation using an exponential window with attenuation factor α (0<α<1) is defined by Equation (10), the estimate values using the window are recursively calculated according to Equation (11).Ea[u(t)]=(1−α)Στu(t−τ)ατ  (10)Ea[u(t)]=αEa[u(t−1)]+(1−α)u(t)  (11)
The dispersion of signals averaged with the window w(t) is Σtwt2 times the dispersion of signals that are not averaged, i.e., it becomes double the area. Therefore, the average power of the estimated error of the correlation value using the exponential window is estimated by Equation (12).
                                                                        E                ⁡                                  [                                      e                    2                                    ]                                            =                            ⁢                                                (                                                            a                      2                                        +                    1                                    )                                ⁢                                                      ∑                    τ                                    ⁢                                                                          ⁢                                                            [                                                                        (                                                      1                            -                            α                                                    )                                                ⁢                                                  α                          τ                                                                    ]                                        2                                                                                                                          =                            ⁢                                                (                                                            a                      2                                        +                    1                                    )                                ⁢                                                      (                                          1                      -                      α                                        )                                    /                                      (                                          1                      +                      α                                        )                                                                                                          (        12        )            
It is found from Equations (9) and (12) that the estimated error with the exponential window having the attenuation factor α matches the error with a rectangular window having a window length N=(1+α)/(1−α). FIG. 9 illustrates the calculation result of the standard deviation of errors for the attenuation factor α of the window. The abscissa represents logarithmic values of 1−α. It is apparent from FIG. 9 that the theoretical values match the experimental values, and that the standard deviation of the estimated error of the correlation value with the exponential window matches the standard deviation of the estimated error of the correlation value with the rectangular window as the squared window having the same area.
However, since the estimation accuracy of the correlation value depends on the area of the squared window, the estimation accuracy of the correlation value is higher as the window length is longer, but it reduces the tracking of variations in sequential processing. FIG. 10 illustrates the results of sequentially estimating a correlation value of two uncorrelated signals with an exponential window. The abscissa represents time, the ordinate represents correlation values, and the line types represent differences in attenuation factor α of the exponential window. It is apparent from FIG. 10 that in the case that the window is long (in case of α=0.998), the absence of correlation can be detected with a high degree of precision but the convergence time becomes long compared to the case that the window is short (in case of α=0.99).
Therefore, it is an object of the present invention to provide a device capable of improving the convergence rate and estimation accuracy in estimating a correlation value.